Does an orbital have its own energy, separate from the electron? If so, is it possible for an electron to have a different energy than the orbital it occupies?
An orbital of current quantum models of atoms and molecules has 3 major interpretations:
orbital(2) is the 3D region describing probability of occurance of an electron, related to orbital(1) . Space regions do not have energy either. (Not exactly true, considering radiation energy and quantum vacuum energy, but nothing comparable to the electron energy at usual conditions.)
orbital(3) is a quantum state of an electron, described by orbital (1) and orbital (2), consisting of set of discrete values of physical quantities as functions of quantum numbers, including energy. In the case this quantum state is not occupied, the orbital energy means "an electron would have this energy".
orbital (4) from the obsolete Bohr model means a path of an electron, which is a crude approximation of the quantum model of an atom.
Last, but not least: Orbitals are not physical things, they are part of the quantum theoretical model of atoms and molecules. Within the given quantum model, an electron cannot have different energy than energy of it's quantum state.
As models of multielectron cases are imperfect, real electron energy usually more or less differ from its energy predicted for the given quantum state.